How change option+key result in macOS Big Sur En-Us keyboard

When I tried enabling the setting (Input source options: Allow a different one for each document), new windows seemed to just use the current input source.

Anyway, you could try running sudo /System/Library/CoreServices/Setup\ Assistant.app/Contents/MacOS/Setup\ Assistant and selecting US as the default input method. (You have to go through the steps of creating a new account, but you can delete it later.) It modifies at least /var/root/Library/Preferences/ByHost/com.apple.HIToolbox.*.plist.

If that doesn't work, try running sudo languagesetup and selecting English, or sudo defaults write /Library/Preferences/.GlobalPreferences.plist AppleLocale en_US and restarting.

You could use a private.xml like this with KeyRemap4MacBook:

<root>
  <vkopenurldef>
    <name>KeyCode::VK_OPEN_URL_KEYBOARDVIEWER</name>
    <url>file:///System/Library/Input%20Methods/KeyboardViewer.app</url>
  </vkopenurldef>
  <vkopenurldef>
    <name>KeyCode::VK_OPEN_URL_QUITKEYBOARDVIEWER</name>
    <url>file:///Applications/QuitKeyboardViewer.app</url>
  </vkopenurldef>
  <item>
    <name>test</name>
    <identifier>test</identifier>
    <autogen>
    __KeyToKey__ KeyCode::OPTION_L,
    ModifierFlag::OPTION_L | ModifierFlag::NONE,
    KeyCode::OPTION_L,
    Option::KEYTOKEY_BEFORE_KEYDOWN,
    KeyCode::VK_OPEN_URL_KEYBOARDVIEWER,
    Option::KEYTOKEY_AFTER_KEYUP,
    KeyCode::VK_OPEN_URL_QUITKEYBOARDVIEWER
    </autogen>
    <autogen>
    __KeyToKey__ KeyCode::OPTION_R,
    ModifierFlag::OPTION_R | ModifierFlag::NONE,
    KeyCode::OPTION_R,
    Option::KEYTOKEY_BEFORE_KEYDOWN,
    KeyCode::VK_OPEN_URL_KEYBOARDVIEWER,
    Option::KEYTOKEY_AFTER_KEYUP,
    KeyCode::VK_OPEN_URL_QUITKEYBOARDVIEWER
    </autogen>
  </item>
</root>

Save this script as /Applications/QuitKeyboardViewer.app in AppleScript Editor:

quit application "KeyboardViewer"

There is a noticeable (but not that long) delay before KeyboardViewer is opened or closed. If for example you hold option and command simultaneously, KeyboardViewer is shown if you press option first but not if you press command first.